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+2 MATHEMATICS slip test question bank

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g‹åbu©lh« tF¥ò - ÁW nj®Î

fâjéaš

m¤Âaha«

6

kÂ¥bg©

édh¡fŸ

10

kÂ¥bg©

édh¡fŸ

ÁW

nj®Îfë‹

v©â¡if

( 6 kÂ¥bg© )

ÁW

nj®Îfë‹

v©â¡if

( 10 kÂ¥bg© )

muR bghJ¤

nj®éš

bgW«

Fiwªjg£r

kÂ¥bg©

1. mâfŸ k‰W«

mâ¡nfhitfë‹ ga‹ghLfŸ

32 - 4 0 12

2. bt¡l® Ïa‰fâj«

- 20 0 4 20

3. fy¥bg©fŸ

32 10 4 2 16

4. gFKiw tot¡ fâj«

- 17 0 4 20

5. tif E©fâj« -

ga‹ghLfŸ I

- - - - -

6. tif E©fâj« -

ga‹ghLfŸ I I

16 11 2 2 16

7. bjhif E©fâj« -

ga‹ghLfŸ

8 9 1 2 10

8. tif¡bfG¢ rk‹ghLfŸ

- 13 0 2 10

9. jåãiy¡ fz¡»aš

24 15 3 3 22

10. ãfœjfÎ¥ gutš

- - - - -

bkh¤j« 112 95 14 19 128


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1. mâfŸ k‰W« mâ¡nfhitfë‹ ga‹ghLfŸ( 6 kÂ¥bg© édh¡fŸ)

ÁW nj®Î 1

1.

53

21A vd;w mzpapd; NrHg;igf; fz;L> . A A ) A adj ( ) A adj (A vd;gijr; rhpghHf;f.

2.

41

21A vdpy;> 2IAAAadjAadjA vd;gjidr; rhpghH.

3.

344

101

334

A -,d; NrHg;G mzp A vd epWTf.

4.

544

434

221

A -f;F> 1 AA vdf; fhl;Lf.

5.

11715

4312

3111

vd;w mzpapd; juk; fhz;f.

6.

323

432

111


7.

3963

2642

1321


8.

1012

7436

3124


ÁW nj®Î 2

1.

2751

5121

1513


2.

0312

0101

0213


3.

0111

1032

1210


4.

7363

2142

3121


5.

6721

3142

4321



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6.

432

323

111


gpd;tUk; rkd;ghLfspd; njhFg;G xUq;fikT cilajh vd;gij Muha;f. xUq;fikT cilajhapd; mtw;iwjjPHf;f

7. 62;1832;7 zyzyxzyx

8. 52687;13283;1474 zyxzyxzyx

ÁW nj®Î 3

1. mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghLfspd; njhFg;Gfisj; jPHf;f : 1664;832 yxyx

2. mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghLfspd; njhFg;Gfisj; jPHf;f : 18108;954 yxyx

3.mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghLfspd; njhFg;Gfisj; jPHf;f :

423;1;522 zyxzyxzyx

4. mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghLfspd; njhFg;Gfisj; jPHf;f :

10633;8422;42 zyxzyxzyx

5. NeHkhW mzpfhzy; Kiwapy; jPHf;f : 832,3 yxyx

6. NeHkhW mzpfhzy; Kiwapy; jPHf;f :

1123,72 yxyx

7. NeHkhW mzpfhzy; Kiwapy; jPHf;f :

02,137 yxyx

8. mzpfspd; NeHkhWfSf;Fhpa thpirkhw;W tpjpapid vOjp ep&gp.

ÁW nj®Î 4

1.

37

25A kw;Wk;

11

12B vdpy; 111

ABAB vd;gjidr; rhpghH.

2.

37

25A kw;Wk;

11

12B vdpy;

TTT

ABAB vd;gjidr; rhpghH.

3.

11

21A kw;Wk;

21

10B vdpy; 111

ABAB .

4.

312

321

111

A -,d; NrHg;G mzpiaf; fhz;f.

5. gpd;tUk; mzpfspd; NeHkhW mzpfisf; fhz;f.

121

022

113

A

6.

gpd;tUk; mzpfspd; NeHkhW mzpfisf; fhz;f

120

031

221

7. .


4110

215

318

8. .


221

131

122

2. bt¡l® Ïa‰fâj« (10 kÂ¥bg© édh¡fŸ )

ÁW nj®Î 5

1. xU Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xNu Gs;spapy; re;jpf;Fk; vd;gjid ntf;lH Kiwapy; epWTf 2. cos A + B = cosA cosB − sinA sinB vd ntf;lH Kiwapy; ep&gp. 3. cos A − B = cosA cosB + sinA sinB vd ntf;lH Kiwapy; ep&gp. 4. sin A − B = sinA cosB − cosA sinB vd ntf;lH Kiwapy; ep&gp.


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5. sin A + B = sinA cosB + cosA sinB vd ntf;lH Kiwapy; ep&gp.

ÁW nj®Î 6

1. 𝑎 = 2 𝑖 + 3𝑗 − 𝑘 , 𝑏 = −2 𝑖 + 5 𝑘 , 𝑐 = 𝑗 − 3 𝑘 vdpy; 𝑎 × 𝑏 × 𝑐 = 𝑎 . 𝑐 𝑏 − 𝑎 . 𝑏 𝑐 vd rhpahHf;f

2. 𝑎 = 𝑖 + 𝑗 + 𝑘 , 𝑏 = 2 𝑖 + 𝑘 , 𝑐 = 2 𝑖 + 𝑗 + 𝑘 , 𝑑 = 𝑖 + 𝑗 + 2 𝑘 vdpy

; 𝑎 × 𝑏 × 𝑐 × 𝑑 = 𝑎 𝑏 𝑑 𝑐 − 𝑎 𝑏 𝑐 𝑑 vd rhpahHf;f

3. 𝑥−1

1 =

𝑦+1

−1 =

𝑧

3kw;Wk;

𝑥−2

1 =

𝑦−1

2 =

−𝑧−1

1vd;w NfhLfs; ntl;bf; nfhs;Sk; vdf; fhl;Lf. NkYk; mit ntl;Lk;

Gs;spiaf; fhz;f

4. 𝑥−1

3 =

𝑦−1

−1 =

𝑧+1

0 kw;Wk;

𝑥−4

2 =

𝑦

0 =

𝑧+1

3vd;w NfhLfs; ntl;Lk; vdf; fhl;b mit ntl;Lk; Gs;spiaf; fhz;f

5. ntl;Lj;Jz;L tbtpy; xU jsj;jpd; rkd;ghl;ilj; jUtpf;f.

ÁW nj®Î 7

1. (2,−1,−3) topNar; nry;yf;$baJk; 𝑥−2

3 =

𝑦−1

2 =

𝑧−3

−4 kw;Wk;

𝑥−1

2 =

𝑦+1

3 =

𝑧−2

2 Mfpa NfhLfSf;F ,izahf

cs;sJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f

2. (1,3,2) vd;w Gs;sp topr; nry;tJk; 𝑥+1

2 =

𝑦+2

−1 =

𝑧+3

3 kw;Wk;

𝑥−2

1 =

𝑦+1

2 =

𝑧+2

2vd;w

NfhLfSf;F,izahdJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f 3. (−1,3,2) vd;w Gs;sp topr; nry;tJk; 𝑥 + 2𝑦 + 2𝑧 = 5 kw;Wk; 3𝑥 + 𝑦 + 2𝑧 = 8 Mfpa jsq;fSf;Fr; nrq;Fj;jhdJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

4. 𝑥−2

2 =

𝑦−2

3 =

𝑧−1

3vd;w Nfhl;il cs;slf;fpaJk;

𝑥+1

3 =

𝑦−1

2 =

𝑧+1

1 vd;w Nfhl;bw;F ,izahdJkhd jsj;jpd;

ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

5. 𝑥−2

2 =

𝑦−2

3 =

𝑧−1

−2vd;w Nfhl;il cs;slf;fpaJk; (−1,1,−1) vd;w Gs;sp topNar; nry;yf; $baJkhd ntf;lH

kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f

ÁW nj®Î 8

1. (−1,1,1) kw;Wk; (1,−1,1) Mfpa Gs;spfs; toNar; nry;yf; $baJk; 𝑥 + 2𝑦 + 2𝑧 = 5 vd;w jsj;jpw;F nrq;Fj;jhf miktJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghl;ilf; fhz;f

2. A ( 1,−2,3) kw;Wk; B (−1,2,−1) vd;w Gs;spfs; topNar; nry;yf;$baJk; 𝑥−2

2 =

𝑦+1

3 =

𝑧−1

4vd;w Nfhl;bw;F

,izahdJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f. 3. (1,2,3) kw;Wk; (2,3,1) vd;w Gs;spfs; topNar; nry;yf; $baJk; 3𝑥 − 2𝑦 + 4𝑧 − 5 = 0 vd;w jsj;jpw;Fr; nrq;Fj;jhfTk; mike;j jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f 4. (2,2,−1) , (3,4,2) kw;Wk; (7,0,6) Mfpa Gs;spfs; topNar; nry;yf;$ba jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghl;ilf; fhz;f

5. 3 𝑖 + 4𝑗 + 2 𝑘 , 2 𝑖 − 2𝑗 − 𝑘 kw;Wk; 7 i + k Mfpatw;iw epiy ntf;lHfshff; nfhz;l Gs;spfs; topNar; nry;Yk; jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f 3. fy¥bg©fŸ (6 kÂ¥bg© édh¡fŸ )

ÁW nj®Î 9

1. i68 -,d; tHf;f%yk; fhz;f.

2. i247 -,d; tHf;f%yk; fhz;f.

3. i32 I xU jPHthff; nfhz;l 03584 24 xxx vDk; rkd;ghl;ilj; jPHf;f.

4. i3 I xU jPHthff; nfhz;l 02032248 234 xxxx vDk; rkd;ghl;bd; jPHTfisf; fhz;f.



7. iBAibaibaiba nn ...2211 vdpy; ep&gp : (i) 222222

22

21

21 ... BAbababa nn

(ii) ZkA

Bk

a

b

a

b

a

b

n

n

,tantan....tantan 11

2

21

1

11

8. iii 74,25,57 kw;Wk; i42 vDk; fyg;ngz;fs; xU ,izfuj;ij mikf;Fk; vd epWTf.

ÁW nj®Î 10

1. MHfd; jsj;jpy; fyg;ngz;fs; ii 42,810 kw;Wk; i3111 mikf;Fk; Kf;Nfhzk; xU nrq;Nfhz Kf;Nfhzk;

vd epWTf.


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2.

iii 3333,33,33 vDk; fyg;ngz;fs; xU rkgf;f Kf;Nfhzj;ij MHfd; jsj;jpy; cUthf;Fk; vd;W

fhl;Lf. 3.

iii 44,1,2 kw;Wk; i53 vDk; fyg;ngz;fs; MHfd; jsj;jpy; xU nrt;tfj;ij cUthf;Fk; vdf; fhl;Lf.

4.

iii 33,73,97 vDk; fyg;ngz;fs; MHfd; jsj;jpy; xU nrq;Nfhz Kf;Nfhzj;ij mikf;Fk; vd epWTf.

5. 21 , ZZ vd;w VNjDk; ,U fyg;ngz;fSf;F (i) 2121 ZZZZ (ii) 2121 argargarg ZZZZ vd ep&gp.

6. 21 , ZZ vd;w VNjDk; ,U fyg;ngz;fSf;F (i)

2

1

2

1

Z

Z

Z

Z (ii) 21

2

1 argargarg ZZZ

Z

vd ep&gp.

7.

cos21

xx vdpy;(i) n

xx

n

n cos21 (ii) ni

xx

n

n sin21 ; Nn vd ep&gp.

8.

sincos;sincos iyix vdpy; nmyx

yxnm

nm cos21

; Nmn , vdf; fhl;Lf.

ÁW nj®Î 11

1. epWTf: 4

cos211 2

2n

ii

n

nn

; Nn

2. epWTf: 3

cos23131 1 nii nnn ; Nn

3. epWTf: 2

cos2cos2sincos1sincos1 1

nii nnnn ; Nn

4. n vd;gJ kpif KO vz; vdpy; 6

cos233 1 nii nnn vd ep&gpf;f.

5. RUf;Ff:

5

4

cossin

sincos

i

i

6. RUf;Ff:

4

3

cossin

sincos

i

i

7. n vd;gJ kpif KO vz; vdpy;

2sin

2cos

cossin1

cossin1nin

i

in

vd ep&gpf;f.

8. Kf;Nfhz rkdpypia vOjp ep&gp.

ÁW nj®Î 12

1. vy;yh kjpg;GfisAk; fhz;f. 318i

2. jPHf;f: (i) 044 x

3. P vDk; Gs;sp fyg;ngz; khwp zIf; Fwpj;jhy; P-,d; epakg;ghijia gpd;tUtdtw;wpw;F fhz;f. 232 z .

4. 13 vdpy 02

1

1

1

21

1

vd epWTf.

5. sinsinsin0coscoscos vdpy; gpd;tUtdtw;iw epWTf:

(i) cos33cos3cos3cos

(ii) sin33sin3sin3sin

6. (iii) 02cos2cos2cos

(iv) 02sin2sin2sin

7. 1z ,d; tPr;R6

kw;Wk; 1z ,d; tPr;R

32

vdpy; 1z vd epWTf.

8. P vDk; Gs;sp fyg;ngz; khwp zIf; Fwpj;jhy; P-,d; epakg;ghijia gpd;tUtdtw;wpw;F fhz;f.

iziz 55

3. fy¥bg©fŸ (10 kÂ¥bg© édh¡fŸ )


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ÁW nj®Î 13

1. 𝑥2 − 2𝑝𝑥 + 𝑝2 + 𝑞2 = 0 vd;w rkd;ghl;bd; %yq;fs; 𝛼,𝛽 kw;Wk; tan θ = 𝑞

𝑦+𝑝 vdpy;

𝑦+𝛼 𝑛 − 𝑦+𝛽 𝑛

𝛼 − 𝛽 = 𝑞𝑛−1 sin 𝑛𝜃

sinn𝜃 vd epWTf.

2. 𝛼,𝛽 vd;git 𝑥2 − 2𝑥 + 2 = 0 -d; %yq;fs; kw;Wk; cot θ = y + 1 vdpy;

𝑦+𝛼 𝑛 − 𝑦+𝛽 𝑛

𝛼 − 𝛽 =

sin nθ

si nn θ vdf; fhl;Lf

3. 𝑥2 − 2𝑥 + 4 = 0 -,d; %yq;fs; 𝛼 kw;Wk; 𝛽 vdpy; 𝛼𝑛 − 𝛽𝑛 = 𝑖2𝑛+1 sin 𝑛𝜋

3 mjpypUe;J α9 − β9 kjpg;ig ngWf

.4. a = cos 2α + i sin 2α, b = cos 2β + i sin 2βkw;Wk; c = cos 2 γ + i sin 2γ vdpy;

i 𝑎𝑏𝑐 + 1

𝑎𝑏𝑐 = 2 cos 𝛼 + 𝛽 + 𝛾 ii

𝑎2𝑏2 + 𝑐2

𝑎𝑏𝑐 = 2 cos 2 𝛼 + 𝛽 − 𝛾 ±Éì ¸¡ðÎ¸

5. 𝑃 vDk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy; 𝑃-,d; epakg;ghijia arg z−1

z+3 =

π

2vd;w fl;Lghl;;bw;F cl;gl;L

fhz;f

ÁW nj®Î 14

1. 1

2 − i

3

2

3

4-d; vy;yh kjpg;GfisAk; fhz;f kw;Wk; mjd; kjpg;Gfspd; ngUf;fw;gyd; 1 vdTk; fhl;Lf

2. jPHf;f: 𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1 = 0. 3. 𝑥9 + 𝑥5 − 𝑥4 − 1 = 0 vd;w rkd;ghl;ilj; jPHf;f 4. 𝑥7 + 𝑥4 + 𝑥3 + 1 = 0 vd;w rkd;ghl;ilj; jPHf;f.

5. − 3 − i 2

3 d; vy;yh kjpg;GfisAk; fhz;f.

SLIP TEST - 15 [10 mark] ÀÌÓ¨È ÅÊÅ¢Âø

1. 5𝑥 + 12𝑦 = 9 ±ýÈ §¿÷ì§¸¡Î «¾¢ÀÃÅ¨ÇÂõ 𝑥2 − 9𝑦2 = 9 -³ò ¦¾¡Î¸¢ÈÐ ±É ¿¢åÀ¢ì¸.

§ÁÖõ ¦¾¡Îõ ÒûÇ¢¨ÂÔõ ¸¡ñ¸.

2. 𝑥 − 𝑦 + 4 = 0±ýÈ §¿÷ì§¸¡Î ¿£ûÅð¼õ 𝑥2 + 3𝑦2 = 12 -³ò ¦¾¡Î¸¢ÈÐ ±É ¿¢åÀ¢ì¸. §ÁÖõ

¦¾¡Îõ ÒûÇ¢¨ÂÔõ ¸¡ñ¸

3. 𝑥 + 2𝑦 − 5 = 0-³ ´Õ ¦¾¡¨Äò ¦¾¡Î§¸¡¼¡ ¸×õ(6,0)ÁüÚõ (−3,0)±ýÈ ÒûÇ¢¸û ÅÆ¢§Â

¦ºøÄìÜÊÂÐÁ¡É ¦ºùÅ¸ «¾¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡Î ¸¡ñ¸.

4..«¾¢ÀÃÅ¨ÇÂò¾¢ý ¨ÁÂõ 2,4 .§ÁÖõ (2,0) ÅÆ¢§Â ¦ºø¸¢ÈÐ. þ¾ý ¦¾¡¨Äò ¦¾¡Î§¸¡Î¸û

𝑥 + 2𝑦 − 12 = 0 ÁüÚõ 𝑥 − 2𝑦 + 8 = 0 ¬¸¢ÂÅüÈ¢üÌ þ¨½Â¡¸ þÕì¸¢ýÈÉ ±É¢ø

«¾¢ÀÃÅ¨ÇÂò¾¢ý ºÁýÀ¡Î ¸¡ñ¸.

SLIP TEST - 16 [10 mark]

1. ´Õ ¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ Å¼õ ÀÃÅ¨ÇÂ ÅÊÅ¢ÖûÇÐ. «¾ý À¡Ãõ ¸¢¨¼Áð¼Á¡¸ º£Ã¡¸ ÀÃÅ¢ÔûÇÐ. «¨¾ò

¾¡íÌõ þÕ àñ¸ÙìÌ þ¨¼§ÂÔûÇ àÃõ 1500 «Ê. ¸õÀ¢ Å¼ò¨¾ ¾¡íÌõ ÒûÇ¢¸û à½¢ø ¾¨ÃÂ¢Ä¢ÕóÐ

200 «Ê ¯ÂÃò¾¢ø «¨ÁóÐûÇÉ. §ÁÖõ ¾¨ÃÂ¢Ä¢ÕóÐ ¸õÀ¢ Å¼ò¾¢ý ¾¡úÅ¡É ÒûÇ¢Â¢ý ¯ÂÃõ 70 «Ê,

¸õÀ¢Å¼õ 122 «Ê ¯ÂÃò¾¢ø ¾¡íÌõ ¸õÀò¾¢üÌ þ¨¼§Â ¯ûÇ ¦ºíÌòÐ ¿£Çõ ¸¡ñ¸.(¾¨ÃìÌ þ¨½Â¡¸)

2. ´Õ ¦¾¡íÌ À¡Äò¾¢ý ¸õÀ¢ Å¼õ ÀÃÅ¨ÇÂ ÅÊÅ¢ÖûÇÐ. «¾ý ¿£Çõ 40Á£ð¼÷ ¬Ìõ. ÅÆ¢ôÀ¡¨¾Â¡ÉÐ ¸õÀ¢

Å¼ò¾¢ý ¸£úÁð¼ô ÒûÇ¢Â¢Ä¢ÕóÐ 5 Á£ð¼÷ ¸£§Æ ¯ûÇÐ. ¸õÀ¢ Å¼ò¨¾ ¾¡íÌõ àñ¸Ç¢ý ¯ÂÃí¸û 55

Á£ð¼÷ ±É¢ø 30 Á£ð¼÷ ¯ÂÃò¾¢ø ¸õÀ¢ Å¼ò¾¢üÌ ´Õ Ð¨½ ¾¡í¸¢ ÜÎ¾Ä¡¸ì ¦¸¡Îì¸ôÀð¼¡ø

«òÐ¨½ò¾¡í¸¢Â¢ý ¿£Çò¨¾ì ¸¡ñ¸.

3. ´Õ ÃÂ¢ø§Å À¡Äò¾¢ý §Áø Å¨Ç× ÀÃÅ¨ÇÂò¾¢ý «¨Áô¨Àì ¦¸¡ñÎûÇÐ. «ó¾ Å¨ÇÅ¢ý «¸Äõ 100

«ÊÂ¡¸×õ «ùÅ¨ÇÅ¢ý ¯îº¢ôÒûÇ¢Â¢ý ¯ÂÃõ À¡Äò¾¢Ä¢ÕóÐ 10 «ÊÂ¡¸ ×õ ¯ûÇÐ ±É¢ø, À¡Äòò¾¢ý

Áò¾¢Â¢Ä¢ ÕóÐ þ¼ôÒÈõ «øÄÐ ÅÄôÒÈõ 10 «Ê àÃò¾¢ø À¡Äò¾¢ý §Áø Å¨Ç× ±ùÅÇ× ¯ÂÃò¾¢ø

þÕìÌõ?

4. ´Õ Å¡ø Å¢ñÁ£ý ¬ÉÐ ÝÃ¢Â¨Éî ÍüÈ¢ ÀÃÅ¨ÇÂô À¡¨¾Â¢ø ¦ºø¸¢ÈÐ ÁüÚõ ÝÃ¢Âý ÀÃÅ¨ÇÂò¾¢ý

ÌÅ¢Âò¾¢ø «¨Á¸¢ÈÐ. Å¡ø Å¢ñÁ£ý ÝÃ¢ÂÉ¢Ä¢ÕóÐ 80 Á¢øÄ¢Âý ¸¢.Á£. ¦¾¡¨ÄÅ¢ø «¨ÁóÐ þÕìÌõ §À¡Ð


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Å¡ø Å¢ñÁ£¨ÉÔõ ÝÃ¢Â¨ÉÔõ þ¨½ìÌõ §¸¡Î «îÍ¼ý 𝜋

3 §¸¡½ò¾¢¨É ²üÀÎòÐÁ¡É¡ø (i) Å¡ø

Å¢ñÁ£É¢ý À¡¨¾Â¢ý ºÁýÀ¡ð¨¼ì ¸¡ñ¸. (ii) Å¡ø Å¢ñÁ£ý ÝÃ¢ÂÛìÌ ±ùÅÇ× «Õ¸¢ø ÅÃÓÊÔõ

±ýÀ¨¾Ôõ ¸¡ñ¸.(À¡¨¾ ÅÄÐÒÈõ ¾¢ÈôÒ¨¼Â¾¡¸ ¦¸¡û¸)

SLIP TEST - 17 [ 10 mark]

1. ´Õ Ã¡ì¦¸ð ¦ÅÊÂ¡ÉÐ ¦¸¡ÙòÐõ §À¡Ð «Ð ´Õ ÀÃÅ¨ÇÂô À¡¨¾Â¢ø ¦ºø¸¢ÈÐ. «¾ý ¯îº ¯ÂÃõ 4

Á£ð¼÷ ³ ±ðÎõ§À¡Ð «Ð ¦¸¡Ùò¾ôÀð¼ þ¼ò¾¢Ä¢ÕóÐ ¸¢¨¼Áð¼ àÃõ 6 Á£ð¼÷ ¦¾¡¨ÄÅ¢ÖûÇÐ. þÚ¾¢Â¡¸

¸¢¨¼Áð¼Á¡¸ 12Á£ ¦¾¡¨ÄÅ¢ø ¾¨Ã¨Â Åó¾¨¼¸¢ÈÐ ±É¢ø ÒÈôÀð¼ þ¼ò¾¢ø ¾¨ÃÔ¼ý ²üÀÎòÐõ

±È¢§¸¡½õ ¸¡ñ¸.

2. ¾¨ÃÁð¼ò¾¢Ä¢ÕóÐ 7.5Á£ ¯ÂÃò¾¢ø ¾¨ÃìÌ þ¨½Â¡¸ ¦À¡Õò¾ôÀð¼ ´Õ ÌÆ¡Â¢Ä¢ÕóÐ ¦ÅÇ¢§ÂÚõ ¿£÷

¾¨Ã¨Âò ¦¾¡Îõ À¡¨¾ ´Õ ÀÃÅ¨ÇÂò¨¾ ²üÀÎòÐ¸¢ÈÐ. §ÁÖõ þó¾ ÀÃÅ¨ÇÂô À¡¨¾Â¢ý Ó¨É ÌÆ¡Â¢ý

Å¡Â¢ø «¨Á¸¢ÈÐ. ÌÆ¡ö Áð¼ò¾¢üÌ 2.5Á£ ¸£§Æ ¿£Ã¢ý À¡öÅ¡ÉÐ ÌÆ¡Â¢ý Ó¨É ÅÆ¢Â¡¸î ¦ºøÖõ ¿¢¨Ä

ÌòÐì§¸¡ðÊüÌ 3 Á£ð¼÷ àÃò¾¢ø ¯ûÇÐ ±É¢ø ÌòÐì §¸¡ðÊÄ¢ÕóÐ ±ùÅÇ× àÃò¾¢üÌ «ôÀ¡ø ¿£Ã¡ÉÐ

¾¨ÃÂ¢ø Å¢Øõ.

3. ´Õ Å¨Ç× «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ. «¾ý «¸Äõ 48 «Ê, ¯ÂÃõ 20 «Ê. ¾¨ÃÂ¢Ä¢ÕóÐ 10 «Ê

¯ÂÃò¾¢ø Å¨ÇÅ¢ý «¸Äõ ±ýÉ?

4. ´Õ À¡Äò¾¢ý Å¨ÇÅ¡ÉÐ «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ. ¸¢¨¼Áð¼ò¾¢ø «¾ý «¸Äõ 40 «ÊÂ¡¸×õ,

¨ÁÂò¾¢Ä¢ÕóÐ «¾ý ¯ÂÃõ 16 «ÊÂ¡¸×õ ¯ûÇÐ ±É¢ø ¨ÁÂò¾¢Ä¢ÕóÐ ÅÄÐ «øÄÐ þ¼ô ÒÈò¾¢ø 9 «Ê

àÃò¾¢ø ¯ûÇ ¾¨ÃôÒûÇ¢Â¢Ä¢ÕóÐ À¡Äò¾¢ý ¯ÂÃõ ±ýÉ?

SLIP TEST - 18 [10 mark]

1. ´Õ Ñ¨Æ× Å¡Â¢Ä¢ý §ÁüÜ¨ÃÂ¡ÉÐ «¨Ã-¿£ûÅð¼ ÅÊÅ¢ø ¯ûÇÐ. þ¾ý «¸Äõ 20 «Ê. ¨ÁÂò¾¢Ä¢ÕóÐ

«¾ý ¯ÂÃõ 18 «Ê ÁüÚõ Àì¸î ÍÅÃ¢¸Ç¢ý ¯ÂÃõ 12 «Ê ±É¢ø ²§¾Ûõ ´Õ Àì¸î ÍÅÃ¢Ä¢ÕóÐ 4 «Ê

àÃò¾¢ø §ÁüÜ¨ÃÂ¢ý ¯ÂÃõ ±ýÉÅ¡¸ þÕìÌõ?

2. ´Õ ¿£ûÅð¼ô À¡¨¾Â¢ý ÌÅ¢Âò¾¢ø âÁ¢ þÕìÌÁ¡Ú ´Õ Ð¨½ì§¸¡û ÍüÈ¢ ÅÕ¸¢ÈÐ. þ¾ý ¨ÁÂò ¦¾¡¨Ä×

¾¸× 1

2 ¬¸×õ âÁ¢ìÌõ Ð¨½ì §¸¡ÙìÌõ þ¨¼ôÀð¼ Á£îº¢Ú àÃõ 400 ¸¢§Ä¡ Á£ð¼÷¸û ¬¸×õ

þÕìÌÁ¡É¡ø Ð¨½ì §¸¡ÙìÌõ âÁ¢ìÌõ þ¨¼ôÀð¼ «¾¢¸Àðº àÃõ ±ýÉ?

3. ÝÃ¢Âý ÌÅ¢Âò¾¢Ä¢ÕìÌÁ¡Ú ¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢Â¨É ´Õ ¿£ûð¼ô À¡¨¾Â¢ø ÍüÈ¢ ÅÕ¸¢ÈÐ. «¾ý «¨Ã

¦¿ð¼îº¢ý ¿£Çõ 36 Á¢øÄ¢Âý ¨Áø¸û ¬¸×õ ¨ÁÂò ¦¾¡¨Ä× ¾¸× 0 ⋅ 206 ¬¸×õ þÕìÌÁ¡Â¢ý (𝑖)

¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ Á¢¸ «Õ¸¡¨ÁÂ¢ø ÅÕõ§À¡Ð ¯ûÇ àÃõ (𝑖𝑖) ¦Á÷ìÌÃ¢ ¸¢Ã¸Á¡ÉÐ ÝÃ¢ÂÛìÌ

Á¢¸ò ¦¾¡¨ÄÅ¢ø þÕìÌõ §À¡Ð ¯ûÇ àÃõ ¬¸¢ÂÅü¨Èì ¸¡ñ¸.

4. ´Õ §¸¡-§¸¡ Å¢¨ÇÂ¡ðÎ Å£Ã÷ Å¢¨ÇÂ¡ðÎô ÀÂ¢üº¢Â¢ý §À¡Ð «ÅÕìÌõ §¸¡-§¸¡ Ìîº¢ì¸ÙìÌõ þ¨¼§ÂÔûÇ

àÃõ ±ô¦À¡ØÐõ 8 Á£ ¬¸ þÕìÌÁ¡Ú ¯½÷¸¢È¡÷. «ùÅ¢Õ Ìîº¢¸ÙìÌ þ¨¼ôÀð¼ àÃõ 6 Á£ ±É¢ø «Å÷

µÎõ À¡¨¾Â¢ý ºÁýÀ¡¨¼ì ¸¡ñ¸.

5. ´Õ ºÁ¾Çò¾¢ý §Áø ¦ºíÌò¾¡¸ «¨ÁóÐûÇ ÍÅÃ¢ý Á£Ð 15Á£ ¿£ÇÓûÇ ²½¢Â¡ÉÐ ¾Çò¾¢¨ÉÔõ

ÍÅüÈ¢¨ÉÔõ ¦¾¡ÎÁ¡Ú ¿¸÷óÐ ¦¸¡ñÎ þÕì¸¢ÈÐ ±É¢ø ²½¢Â¢ý ¸£úÁð¼ Ó¨ÉÂ¢Ä¢ÕóÐ 6Á£ àÃò¾¢ø

²½¢Â¢ø «¨ÁóÐûÇ 𝑃 ±ýÈ ÒûÇ¢Â¢ý ¿¢ÂÁôÀ¡¨¾ì ¸¡ñ¸.

6. tif E©fâj« - ga‹ghLfŸ I I (6 kÂ¥bg© édh¡fŸ ) ÁW nj®Î 19

1. zyxu tantantanlog vdpy; 22sin

x

ux vd ep&gp.

2. xzzyyxU vdpy; 0 zyx UUU vdf; fhl;Lf.

3. 2xyez vd;w rhHgpy; tx 2 kw;Wk; ty 1 vDkhW ,Ug;gpd;

dt

dzfhz;f.

4.

22 zyxw vd;w rhHgpy; tztytx ;sin;cos vdpy;dt

dwfhz;f.

5.

zxyw vd;w rhHgpy; tztytx ;sin;cos vdpy;dt

dwfhz;f.

6. byaxzeV kw;Wk; z MdJ x,y-,y; n-Mk; gb rkg;gbj;jhd rhHghapd; Vnbyaxy

Vy

x

Vx

vd epWTf.


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7. xU jdp Crypd; ePsk; kw;Wk; KO miyT Neuk; T vdpy; kT (k vd;gJ khwpyp). jdpCrypd; ePsk; 32.1

nr.kP ,ypUe;J 32.0 nr.kPf;F khWk; NghJ> Neuj;jpy; Vw;gLk; rjtPjg; gpioia fzf;fpLf.

8. u vd;gJ gb n nfhz;l rkg;gbj;jhd x,y ,y; mike;j rhHG vdpy;> y

un

y

uy

yx

ux

1

2

22

vdffhl;Lf.

ÁW nj®Î 20

1. tifaPLfisg; gad;gLj;jp 3 65 Njhuha kjpg;Gfisf; fhz;f.

2. tifaPLfisg; gad;gLj;jp 1.36 Njhuha kjpg;Gfisf; fhz;f.

3. tifaPLfisg; gad;gLj;jp 1.10

1Njhuha kjpg;Gfisf; fhz;f.

4. tifaPLfisg; gad;gLj;jp 697.1 Njhuha kjpg;Gfisf; fhz;f.

5. sin,cos ryrx vd;W ,Uf;FkhW 22log yxw vd tiuaWf;fg;gLfpwJ vdpy;r

w

kw;Wk;

w

-If; fhz;f.

6.

uvyvux 2,22 vd;W ,Uf;FkhWu

w

kw;Wk;

v

w

if

22 yxw -If; fhz;f.

7.

y

xxyu sin2 vdpy; u

y

uy

x

ux 3

vdf; fhl;Lf.

8.

22 yx

xw

vd;w rhHgpy

tytx sin;cos vdpy;

dt

dwfhz;f.

6. tif E©fâj« - ga‹ghLfŸ I I (10 kÂ¥bg© édh¡fŸ ) ÁW nj®Î 21

1. 𝑦 = 𝑥3 + 1vd;fpw tistiuia tiuf.

2. 𝑦2 = 2𝑥3vd;w tistiuia tiuf. 3. 𝑦 = 𝑥3vd;fpw tistiuia tiuf

4. 𝑓 𝑥, 𝑦 = 1

𝑥2 + 𝑦2f;F A+yhpd; Njw;wj;ij rhpghHf;f

5. 𝑢 = sin−1 𝑥−𝑦

𝑥+ 𝑦 vdpy; A+yhpd; Njw;wj;ijg; gad;gLj;jp 𝑥

∂𝑢

∂𝑥 + 𝑦

∂𝑢

∂𝑦 =

1

2 tan𝑢vdf; fhl;Lf

6. 𝑢 = tan−1 𝑥3 + 𝑦3

𝑥−𝑦 vdpy;vdpy; A+yhpd; Njw;wj;ijg; gad;gLj;j𝑥

∂𝑢

∂𝑥 + 𝑦

∂𝑢

∂𝑦 = sin2𝑢vd ep&gpf;f

ÁW nj®Î 22

1. tifaPLfisg; gad;gLj;jp 𝑦 = 1.023

+ 1.024

d; Njhuha kjpg;igf; fzf;fpLf.

2. 𝑤 = 𝑢2𝑒𝑣vd;w rhHgpy;𝑢 = 𝑥

𝑦kw;Wk;𝑣 = 𝑦 log 𝑥vDkhW ,Ug;gpd;

x

w

kw;Wk;

∂𝑤

∂𝑦fhz;f.

3. 𝑢 = 𝑥

𝑦2 − 𝑦

𝑥2vdpy; ∂2𝑢

∂𝑥 ∂𝑦 =

∂2𝑢

∂𝑦 ∂𝑥vd;gij rhpghHf;f.

4. 𝑢 = sin 3𝑥 cos 4𝑦 vdpy; ∂2𝑢

∂𝑥 ∂𝑦 =

∂2𝑢

∂𝑦 ∂𝑥vd;gij rhpghHf;f

5. 𝑢 = tan−1 𝑥

𝑦 vdpy;

∂2𝑢

∂𝑥 ∂𝑦 =

∂2𝑢

∂𝑦 ∂𝑥vd;gij rhpghHf;f

7. bjhif E©fâj« - ga‹ghLfŸ ( 6kÂ¥bg© édh¡fŸ )

ÁW nj®Î 23

1. kjpg;gpLf: dxx

2

0

tanlog

2. kjpg;gpLf:

3

6 cot1

x

dx


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3. . kjpg;gpLf:

3

0 3 xx

dxx

4. . kjpg;gpLf:

3

6 tan1

x

dx

5. kjpg;gpLf:. dxx

1

0

11

log

6. kjpg;gpLf: dxxx

n

1

0

1

7. kjpg;gpLf: dxx

4sin

2

0

9

8. kjpg;gpLf: dxxx 22

0

4 cossin

7. bjhif E©fâj« - ga‹ghLfŸ ( 10 kÂ¥bg© édh¡fŸ )

ÁW nj®Î 24

1. Muk; ‘ ‘ Fj;Jauk; ‘ ℎ ‘ cila $k;gpd; fdmsitf; fhz;f.

2. 𝑥

𝑎

2 3

+ 𝑦

𝑎

2 3

= 1 vd;w tistiuapd; ePsj;ijf; fhz;f

3. Muk; 𝑎cila tl;lj;jpd; Rw;wsitf; fhz;f. 4. 𝑥 = 𝑎 𝑡 − sin 𝑡 , 𝑦 = 𝑎 1 − cos 𝑡 vd;w tistiuapd; ePsj;jpid 𝑡 = 0 Kjy; π tiu fzf;fpLf. 5. 4𝑦2 = 𝑥3vd;w tistiuapy; 𝑥 = 0,ypUe;J𝑥 = 1tiuAs;s tpy;ypd; ePsj;ijf; fhz;f

ÁW nj®Î 25

1. 𝑦 = sin 𝑥vd;w tistiu 𝑥 = 0 , 𝑥 = πkw;Wk; 𝑥-mr;R Mfpatw;why; Vw;gLk; gug;gpid 𝑥-mr;rpidg;

nghWj;J Row;Wk; NghJ fpilf;Fk; jplg;nghUspd; tisgug;G 2π 2 + log 1 + 2 vd epWTf

2. 𝑥 = 𝑎 𝑡 + sin 𝑡 , 𝑦 = 𝑎 1 + cos 𝑡 vd;w tl;l cUs;tis mjd; mbg;gf;fj;ijg; (𝑥 −mr;R) nghWj;J Row;Wtjhy; Vw;gLk; jplg; nghUspd; tisg;gug;igf; fhz;f

3. 𝑦2 = 4𝑎𝑥 vd;w gutisaj;jpd; mjd; nrt;tfyk; tiuapyhd gug;gpid 𝑥 −mr;rpd; kPJ Row;Wk;NghJ fpilf;Fk; jplg; nghUspd; tisgug;igf; fhz;f 4. Muk; 𝑟myFfs;cs;s Nfhsj;jpd; ikaj;jpypUe;J 𝑎kw;Wk;𝑏myFfs; njhiytpy; mike;j ,U ,izahd jsq;fs; Nfhsj;ij ntl;Lk;NghJ ,ilg;gLk; gFjpapd; tisgug;G 2π 𝑟 𝑏 − 𝑎 vd epWTf. ,jpypUe;J Nfhsj;jpd; tisgug;ig tUtp. 𝑏 > 𝑎 8. tif¡bfG¢ rk‹ghLfŸ ( 10 kÂ¥bg© édh¡fŸ )

ÁW nj®Î 26

1. xU ,urhad tpistpy;> xU nghUs; khw;wk; milAk; khW tPjkhdJ 𝑡 Neuj;jpy; khw;wkilahj mg;nghUspd; mstpw;F tpfpjkhf cs;sJ. xU kzp Neu Kbtpy; 60 fpuhKk; kw;Wk; 4 kzp Neu Kbtpy; 21 fpuhKk; kPjkpUe;jhy;> Muk;g epiyapy;> mg;nghUspd; vilapidf; fhz;f. vj;jid fpuhk; ,Ue;jpUf;Fk; 2. Ez;ZapHfspd; ngUf;fj;jpy;> ghf;Bhpahtpd; ngUf;ftPjkhdJ mjpy; fhzg;gLk; ghf;Bhpahtpd; vz;zpf;iff;F tpfpjkhf mike;Js;sJ. ,g;ngUf;fj;jhy; ghf;Bhpahtpd; vz;zpf;if 1 kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp Neu Kbtpy; ghf;Bhpahtpd; vz;zpf;if Muk;g epiyiaf; fhl;bYk ;35 klq;fhFk; vdf; fhl;Lf 3. xU efuj;jpy; cs;s kf;fs; njhifapd; tsHr;rptPjk; me;Neuj;jpy; cs;s kf;fs; njhiff;F tpfpjkhf mike;Js;sJ. 1960 Mk; Mz;by; kf;fs; njhif 130000 vdTk; 1990 ,y; kf;fs; njhif 160000 MfTk; ,Ug;gpd; 2020 Mk; Mz;by; kf;fs; njhif vt;tsthf ,Uf;Fk;? 4. xU fjphpaf;fg; nghUs; rpijAk; khWtPjkhdJ mjd; vilf;F tpfpjkhf mike;Js;sJ. mjd; vil 10 kp.fpuhk; Mf ,Uf;Fk; NghJ rpijAk; khWtPjk; ehnshd;Wf;F 0.051 kp.fpuhk; vdpy; mjd; vil 10 fpuhkpypUe;J 5 fpuhkhff; Fiw vLj;Jf; nfhs;Sk; fhy msitf; fhz;f


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5. xU tq;fpahdJ njhlH $l;L tl;b Kiwapy; tl;biaf; fzf;fpLwfpwJ. mjhtJ tl;b tPjj;ij me;je;j Neuj;jpy; mrypd; khW tPjj;jpy; fzf;fpLfpwJ. xUtuJ tq;fp ,Ug;gpd; njhlHr;rpahd $l;L tl;b %yk; Mz;nlhd;Wf;F 8%tl;b ngUfpwJ vdpy;> mtuJ tq;fpapUg;gpd; xU tUl fhy mjpfhpg;gpd; rjtPjj;ijf; fzf;fpLf. [𝑒 .08 = 1.0833vLj;Jf; nfhs;f 6. &. 1000 vd;w njhiff;F njhlHr;rp $l;L tl;b fzf;fplg;gLfpwJ. tl;b tPjk; Mz;nlhd;Wf;F 4 rjtPjkhf ,Ug;gpd;> mj;njhif vj;jid Mz;Lfspy; Muk;gj; njhifiag; Nghy; ,U klq;fhFk;? ( log𝑒2 = 0.6931)

7. Nubak; rpijAk; khWtPjkhdJ> mjpy; fhzg;gLk; mstpw;F tpfpjkhf mike;Js;sJ. 50 tUlq;fspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpwJ vdpy; 100 tUl Kbtpy; kPjpapUf;Fk; msT vd;d? [𝐴0I Muk;g msT vdf; nfhs;f].

ÁW nj®Î 27

1. ntg;gepiy 15° 𝐶cs;s xU miwapy; itf;fg;gl;Ls;s NjePhpd; ntg;gepiy 100° 𝐶MFk;. mJ 5 epkplq;fspy;60°𝐶Mf Fiwe;J tpLfpwJ. NkYk; 5 epkplk; fopj;J NjePhpd; ntg;g epiyapid fhz;f 2. xU ,we;jtH cliy kUj;JtH ghpNrhjpf;Fk; NghJ> ,we;j Neuj;ij Njhuhakhf fzf;fpl Ntz;bAs;sJ. ,we;jthpd; clypd; ntg;gepiy fhiy 10.00 kzpastpy; 93.4 ° 𝐹vd Fwpj;Jf; nfhs;fpwhH. NkYk; 2 kzp Neuk; fopj;J ntg;gepiy msit 91.4° 𝐹 vdf; fhz;fpwhH. miwapd; ntg;gepiy msT (epiyahdJ)72° 𝐹 vdpy;> ,we;j Neuj;ijf; fzf;fpLf. (xU kdpj clypd;

rhjhuz c\;z epiy 98.6° 𝐹vdf; nfhs;f). log𝑒19.4

21.4 = − 0.0426 , log𝑒

26.6

21.4 = − 0.00945

3. 𝑑2𝑦

𝑑𝑥2 − 3𝑑𝑦

𝑑𝑥 + 2𝑦 = 2𝑒3𝑥vd;w tiff;nfOr; rkd;ghl;ilj; jPHf;f. ,q;F 𝑥 = log2 vdpy; 𝑦 = 0 kw;Wk; 𝑥 = 0 vdpy; 𝑦 = 0

4. jPHf;f : 𝐷2 − 6𝐷 + 9 𝑦 = 𝑥 + 𝑒2𝑥 5. jPHf;f : 𝐷2 − 1 𝑦 = cos 2 𝑥 − 2 sin 2𝑥 6. xU Nehahspapd; rpWePhpypUe;J Ntjpg;nghUs; ntspNaWk; mstpid njhlHr;rpahf Nfj;NjlH vd;w fUtpapd; %yk; fz;fhzpf;fg;gLfpwJ. 𝑡 = 0

vd;w Neuj;jpy; Nehahspf;F 10 kp.fpuhk; Ntjpg;nghUs; nfhLf;fg;gLfpwJ. ,J −3𝑡1 2 kp.fpuhk; / kzp vd;Dk; tPjjj;jpy; ntspNaWfpwJ vdpy;> (i) Neuk;𝑡 > 0vDk;NghJ> Nehahspapd; clypYs;s Ntjpg;nghUspd; msitf; fhZk; nghJr; rkd;ghL vd;d? (ii) KOikahf Ntjpg;nghUs; ntspNaw vLj;Jf; nfhs;Sk; Fiwe;jgl;r fhy msT vd;d?

9. jåãiy¡ fz¡»aš ( 6 kÂ¥bg© édh¡fŸ )

ÁW nj®Î 28

(1) qpqp ~ vdf; fhl;Lf.

(2) pqqpqp vdf; fhl;Lf.

(3) pqqpqp ~~ vdf; fhl;Lf.

(4) qpqp ~~~ vdf; fhl;Lf.

(5) qp kw;Wk; pq rkhdkw;wit vdf; fhl;Lf.

(6) qpqp vd;gJ xU nka;ik vdf; fhl;Lf.

(7) rqp ~ -f;Fhpa nka; ml;ltizia mikf;f.

(8) rqp -,d; nka; ml;ltizia mikf;f.

ÁW nj®Î 29

1. rqp -,d; nka; ml;ltizia mikf;f.

2. qpqp ~ -,d; nka; ml;ltizia mikf;f.

3. rqp -,d; nka; ml;ltizia mikf;f.

4. qPqp ~~~ vdf; fhl;Lf.

5. ~ ~p q p xU nka;ik vdf; fhl;Lf.

6. qpq ~ xU Kuz;ghL vdf; fhl;Lf.

7. qpqP ~~ xU nka;ikah vd;gjid nka; ml;ltiziaf; nfhz;L jPHkhdpf;f.

8. pqpp ~~ xU nka;ikah vd;gjid nka; ml;ltiziaf; nfhz;L jPHkhdpf;f.

ÁW nj®Î 30


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1. ,Z xU Kbtw;w vgPypad; Fyk; vd epWTf

2.

10

01,

10

01,

10

01,

10

01Mfpa ehd;F mzpfSk; mlq;fpa fzk; mzpg;ngUf;fypd;

fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf. 3. Fyj;jpd; ePf;fy; tpjpfis vOjp epWTf. 4. vjpHkiw tpjpapid vOjp ep&gp. 5. 1 ,d; 4-Mk; gb %yq;fs; ngUf;fypd; fPo; vgPypad; Fyj;ij mikf;Fk; vd epWTf 6. 1 ,d; 3 Mk; gb %yq;fs; xU Kbthd vgPypad; Fyj;ij ngUf;fypd; fPo; mikf;Fk; vdf; fhl;Lf. 7. 2 X 2 thpir nfhz;l G+r;rpakw;w Nfhit mzpfs; ahTk; Kbtw;w vgPypad; my;yhj Fyj;ij mzp ngUf;fypd; fPo; mikf;Fk; vdf; fhl;Lf. (,q;F mzpapd; cWg;Gfs; ahTk; RIr; NrHe;jit). 8. G+r;rpakw;w fyg;ngz;fspd; fzk;> fyg;ngz;fspd; tof;fkhd ngUf;fypd; fPo; xU vgPypad; Fyk; vdf; fhl;Lf. 9. jåãiy¡ fz¡»aš ( 10 kÂ¥bg© édh¡fŸ )

ÁW nj®Î 31

1. 1 00 1

, ω 00 ω2 , ω

2 00 ω

, 0 11 0

, 0 ω2

ω 0 ,

0 ωω2 0

vd;fpw fzk; mzpg;ngUf;fypd; fPo;

xU Fyj;ij mikf;Fk; vdf; fhl;Lf.(ω3 = 1) 2. 11-,d; kl;Lf;F fhzg;ngw;w ngUf;fypd;fPo; 1 , 3 , 4 , 5 , 9 vd;w fzk; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf. 3. 𝑍7 − 0 , ⋅ x7 xU Fyj;ij mikf;Fk; vdf; fhl;Lf

4. G+r;rpakw;w fyg;ngz;fspd; fzkhd 𝐶 − 0 ,y; tiuaWf;fg;gl;l 𝑓1 𝑧 = 𝑧, 𝑓2 𝑧 = − 𝑧, 𝑓3 𝑧 = 1

𝑧, 𝑓4 𝑧 =

− 1

𝑧 ∀ 𝑧 ∈ 𝐶 − 0 vd;w rhHGfs; ahTk; mlq;fpa fzk;𝑓1 , 𝑓2 , 𝑓3 , 𝑓4MdJ

rhHGfspd; NrHg;gpd; fPo; xU vgPypad; Fyk; mikf;Fk; vd epWTf

5. 𝑥 𝑥𝑥 𝑥

; 𝑥 ∈ ℝ − 0 vd;w mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa fzk; 𝐺MdJ

mzpg;ngUf;fypd; fPo; xU Fyk; vdf; fhl;Lf.

ÁW nj®Î 32

1. 𝑎 00 0

, 𝑎 ∈ ℝ − 0 mikg;gpy; cs;s vy;yh mzpfSk; mlq;fpa fzk; mzpg;ngUf;fypd; fPo;

xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf. 2. ℤ , ∗ xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf. ,q;F ∗MdJ 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2 vDkhW tiuaWf;fg;gl;Ls;sJ.

3. 𝐺vd;gJ kpif tpfpjKW vz; fzk; vd;f.𝑎 ∗ 𝑏 = 𝑎𝑏

3,𝑎 , 𝑏 ∈ 𝐺vDkhW tiuaWf;fg;gl;l nrayp ∗-

,d; fPo; 𝐺 xU Fyj;ij mikf;Fk; vdf; fhl;Lf. 4. 1Ij; jtpu kw;w vy;yh tpfpjKW vz;fSk; mlq;fpa fzk; 𝐺vd;f. 𝐺y; ∗I𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏, ∀ 𝑎 , 𝑏 ∈ 𝐺vDkhW tiuaWg;Nghk;. ( 𝐺 ,∗ )xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf 5. −1 –I jtpu kw;w vy;yh tpfpjKW vz;fSk; cs;slf;fpa fzk; 𝐺MdJ 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏 𝑎 , 𝑏 ∈ 𝐺.vDkhW tiuaWf;fg;gl;l nrayp ∗ - ,d; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf.

ÁW nj®Î 33

1. tof;fkhd ngUf;fypd; fPo; 1 −,d; 𝑛 −Mk; gb %yq;fs; Kbthd Fyj;ij mikf;Fk; vdf; fhl;Lf 2.. 𝑍𝑛 , +𝑛 xU Fyk; vdf; fhl;Lf 3. 𝐺 = 2𝑛 / 𝑛 ∈ ℤ vd;w fzkhdJ ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf 4. 𝑧 = 1vDkhW cs;s fyg;ngz;fs; ahTk; mlq;fpa fzk; 𝑀MdJ fyg;ngz;fspd; ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf; fhl;Lf.

5. 𝐺 = 𝑎 + 𝑏 2 / 𝑎, 𝑏 ∈ 𝑄 vd;gJ $l;liyg; nghWj;J xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf.

ÁW nj®Î 34

1) xU rktha;g;G khwp 𝑋-,d; epfo;jfT epiwr;rhHG guty; gpd;tUkhW cs;sJ

𝑋 0 1 2 3 4 5 6

𝑃 𝑋 = 𝑥 𝑘 3𝑘 5𝑘 7𝑘 9𝑘 11𝑘 13𝑘

1 𝑘-,d; kjpg;G fhz;f. (2) 𝑃 𝑋 < 4 , 𝑃 𝑋 ≥ 5 , 𝑃 3 < 𝑋 ≤ 6 ,tw;wpd; kjpg;G fhz;f.

(3) 𝑃 𝑋 ≤ 𝑥 >1

2Mf ,Uf;f 𝑥,d; kPr;rpW kjpg;G fhz;f.

2) xU nfhs;fyj;jpy; 4 nts;is kw;Wk; 3 rptg;Gg; ge;JfSk; cs;sd. 3 ge;Jfis xt;nthd;whf vLf;Fk;NghJ>

rptg;G epwg; ge;Jfspd; vz;zpf;ifapd; epfo;jfTg; guty; (epiwr;rhHG) fhz;f.


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3) Ie;J taJila xU caHe;j tif ehapd; KO Mal;fhyk; xU rktha;g;G khwpahFk;. mjd; guty; rhHG

(NrHg;G) 𝐹 𝑥 = 0 , 𝑥 ≤ 5

1 − 25

𝑥2 , 𝑥 > 5 vdpy; 5 taJila eha;(i) 10Mz;LfSf;Ff; Fiwthf(ii) 8Mz;LfSf;Ff;

Fiwthf(iii) 12,ypUe;J15Mz;Lfs; tiu capH tho;tjw;fhd epfo;jfT fhz;f. mjpfkhf 4) xU rktha;g;G khwp𝑥,d; epfo;jfT mlHj;jpr; rhHG

5) 𝑓 𝑥 = 𝑘xα − 1e− β xα , 𝑥 ,α , β > 0

0 , k‰wgo vdpy;(i) 𝑘 ,d; kjpg;G fhz;f.(ii) 𝑃 ( 𝑋 > 10 )fhz;f.

5) xU NgUe;J epiyaj;jpy;> xU epkplj;jpw;F cs;Ns tUk; NgUe;Jfspd; vz;zpf;if gha;]hd; gutiyg; ngw;wpUf;fpwJ vdpy;λ = 0.9 ,vdf; nfhz;L

i. 5 epkpl fhy ,ilntspapy; rhpahf 9 NgUe;Jfs; cs;Ns tu ii. 8 epkpl fhy ,ilntspapy; 10f;Fk; Fiwthf NgUe;Jfs; cs;Ns tu

iii. 11 epkpl fhy ,ilntspapy; Fiwe;jgl;rk; 14 NgUe;Jfs; cs;Ns tu epfo;jfT fhz;f. 6) xU efuj;jpy; thlif tz;b Xl;LdHfshy; Vw;gLk; tpgj;Jfspd; vz;zpf;if gha;]hd; gutiy

xj;jpUf;fpwJ.,jd; gz;gsit 3 vdpy;>1000 Xl;LeHfspy;(i)xU tUlj;jpy; xU tpgj;Jk; Vw;glhky; (ii)xU tUlj;jpy; %d;W tpgj;JfSf;F Nky; Vw;glhky; ,Uf;Fk;gbahd Xl;LdHfspd; vz;zpf;ifiaf; fhz;f. 𝑒−3 = 0.0498

ÁW nj®Î 35

1. xU ,ay;epiyg; gutypd; epfo;jfTg; guty;𝑓 𝑥 = 𝑐𝑒− x2 + 3𝑥 , − ∞ < 𝑋 < ∞vdpy; 𝑐 , 𝜇 ,𝜎2,tw;iwf; fhz;f 2. ,ay;epiy khwp𝑋-d; ruhrhp 6 kw;Wk; jpl;l tpyf;fk; 5 MFk;. i 𝑃 0 ≤ 𝑋 ≤ 8 ii 𝑃 𝑋 − 6 < 10 Mfpatw;iwf; fhz;f 3. xU NjHtpy; 1000 khztHfspd; ruhrhp kjpg;ngz; 34 kw;Wk; jpl;l tpyf;fk; 16 MFk;. kjpg;ngz; ,ay;epiyg; gutiy ngw;wpUg;gpd; (i) 30 ,ypUe;J 60 kjpg;ngz;fSf;fpilNa kjpg;ngz; ngw;w khztHfspd; vz;zpf;if (ii)kj;jpa 70%khztHfs; ngWk; kjpg;ngz;fspd; vy;iyfs; ,tw;iwf; fhz;f. 4. ,ay;epiyg; gutypd; epfo;jfT mlHj;jpr; rhHG𝑓 𝑥 = 𝑘e− 2 x2 + 4𝑥 , − ∞ < 𝑋 < ∞vdpy; 𝑘 , 𝜇kw;Wk;σ2,d; kjpg;G fhz;f 5. etPd rpw;We;Jfspy; nghUj;jg;gLk; rf;fuq;fspypUe;J rktha;g;G Kiwapy; NjHe;njLf;fg;gLk; rf;fuj;jpd; fhw;wOj;jk; ,ay;epiyg; gutiy xj;jpUf;fpwJ. fhw;wOj;j ruhrhp 31 psi. NkYk; jpl;l tpyf;fk; 0.2 psi vdpy; i a 30.5 psi f;Fk 31.5 𝑝𝑠𝑖 f;Fk; ,ilg;gl;l fhw;wOj;jk;(b) 30psif;Fk;32 psif;Fk; ,ilg;gl;l fhw;wOj;jk; vd ,Uf;Fk;gbahf rf;fuj;jpid NjHe;njLf;f epfo;jfT fhz;f. (ii)rktha;g;G Kiwapy; NjHe;njLf;fg;gLk; rf;fuj;jpd; fhw;wOj;jk; 30.5 psif;F mjpfkhf ,Uf;f epfo;jfT fhz;f. 6. xU Fwpg;gpl;l fy;Y}hpapy; 500 khztHfspd; vilfs; xU ,ay;epiyg; gutiy xj;jpUg;gjhff nfhs;sg;gLfpwJ . ,jd; ruhrhp 151 gTz;LfshfTk; jpl;l tpyf;fk; 15 gTz;LfshfTk; cs;sd. (i) 120gTz;Lf;Fk;155gTz;Lf;Fk; ,ilNaAs;s khztHfs;(ii) 185gTz;Lf;F Nky; epiwAs;s khztHfspd; vz;zpf;if fhz;f

𝒛 0 ⋅ 267 2 ⋅ 067 2 ⋅ 27 2 ⋅ 5 0 ⋅ 4 1 ⋅ 2 2 0 ⋅ 25 1 ⋅ 63 1 ⋅ 4

gug;G 0 ⋅ 1064 0 ⋅ 4808 0 ⋅ 4884 0 ⋅ 4938 0 ⋅ 1554 0 ⋅ 3849 0 ⋅ 4772 0 ⋅ 0987 0 ⋅ 4484 0 ⋅ 35

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